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Hartree–Fock Approximate Molecular Orbital Theory Justin T. Fermann 3 lectures on theory 1 lecture on programming Purpose First, we define the problem, beginning with the Schr¨odinger equation ˆ = EΨ. HΨ (1) Our goal is to come up with an analytic equation for the energy which can be minimized with respect to some variational parameter(s) to give an upper bound on the energy. To do this, we must ˆ 1. Understand the hamiltonian operater H. 2. Find an appropriate wave function Ψ which allows simple calculation of the electronic energy. 3. Examine potential variational parameters to figure out how to minimize the energy. 1 It will likely be convenient to have an outline of where we’re going, so here it is: I. Puropse (which we’ve been through) ˆ II. The Hamiltonian, H • Kinetic energy operators • Coulombic potential operators III. The Wave Function, Ψ • 1 electron orbitals • Hartree products • Slater determinants IV. Hamiltonian as Energy Operator, Schr¨ odinger Equation ˆ • E = hΨ|H|Ψi • Integrals over one electron operators • Integrals over two electron operators • Specific case energy expressions • General form of HF energy V. What Variational Parameter? • LCAO–MO theory, energy in AO basis • Density Matrices VI. Hartree–Fock Equations • Lagrangs’s Undetermined Multipliers • A load ’o math VII. Matrix Formalism VIII. Program Outline 2 ˆ The Hamiltonian, H The Hamiltonian is the total energy operator for a system, and is written as the sum of the kinetic energy of all the components of the system and the internal potential energy. In an atom or molecule, comprised of positive nuclei and negative electrons, the potential energy is simply that due to the coulombic interactions present. Thus for the kinetic energy in a system of M nuclei and N electrons: TˆN = − Tˆe = −